Ultimately, the approach to physical reduction that I advocate in this disserta- tion can be classified as a dynamical systems approach, which is neither wholly systematic nor wholly piecemeal, but based instead on the use of reduction tem- plates. In deducing an image modelMh∗for some modelMhofThfrom a model
MlofTl, a complete proof may be difficult or practically impossible to come by in the case of highly complicated systems. As we will see in the next chapter, in such a case plausible but unproven assumptions about the system in question need to be made, and the deduction of Mh∗ within a particular domain of Ml should be given in the form of a template rather than a complete proof. For simpler systems, it may be possible to provide templates that are sufficiently complete that they constitute complete proofs of the image model.
In summary, the goal of template-based reduction as much to tell a story about the relationship between the alternative descriptions of the physical world
accuracy of the high-level theory’s models on the basis of the low-level theory’s models (the story, if told in sufficient detail, should amount to such a proof). Necessarily, given the semantic, dynamical systems approach taken here, as well as the specialisation to theories in physics, it will be a story told in the language of mathematics.
Chapter 2
The Classical Domain of
Non-Relativistic Quantum
Mechanics
The first two theories that I consider as an application of dynamical systems, template-based reduction are Newtonian mechanics and nonrelativistic quantum mechanics. Specifically, I consider the reduction of a class of models of Newto- nian mechanics describing N macroscopic centers of mass interacting through a time-independent potential, to a class of corresponding models within Ev- erettian and Bohmian quantum mechanics. Throughout, I refer to the Everett theory as the Bare/Everett theory as a reminder that, from a mathematical point of view, Everett’s theory just is the bare formalism of quantum theory, prescribing unitary dynamics for a vector in a Hilbert space without collapse. I do not address more abstract algebraic approaches to quantum theory in this thesis; the interested reader can consult, for example, Landsman’s [63], or [64]. In section 2.1, I discuss the measurement problem as it relates to the re- duction of classical to quantum mechanics, and my reasons for considering the
Newtonian mechanics to be reduced, and the models of Everettian and Bohmian quantum mechanics to which I reduce it. In section 2.3, I describe the basic mechanisms of decoherence, measurement and effective wave function collapse in Bohmian and Everettian quantum mechanics, including a review of the deco- herent histories framework in the Schrodinger picture. In section 2.4, I provide a template for the DS reduction of the model of Newtonian mechanics to the corresponding model in the Bare/Everett theory. Finally, in section 2.5, I pro- vide a template for the DS reduction of the model of Newtonian mechanics to the corresponding model in the Bohm theory.
2.1
The Measurement Problem
Given the realist background of this thesis, any attempt to reduce classical to quantum theory must take account of the quantum measurement problem. For, it is only through some resolution to this problem that the connection between the microscopic indeterminacy (relative to familiar variables such as position and momentum) of quantum theory and the apparent macroscopic determinacy, and determinism, of classical theory can be elucidated according to the demands of the realist. It is for this reason that, in considering the reductions that I do, I have adopted two proposed resolutions to the measurement problem, the Everett and Bohm theories.
However, I hope that my analysis will contain points of interest for readers skeptical of the Everett and Bohm theories, or of realist approaches to quantum theory more generally. As Wallace has argued at length, Everett’s theory, from a mathematical point of view, is just the bare formalism of quantum theory without collapse [110]. The distinction between the two, as far as usage goes, comes from the added points of metaphysical and epistemological interpretation that the Everett theory attaches to the bare formalism. The most controversial claim of Everett’s theory, that each branch of the total wave function, as defined by the requirement of decoherence, corresponds to an independent ‘world’ with
its own determinate reality, emerges as a consequence of taking the theory’s mathematics seriously as a guide to the structure of the physical world; ‘taking the math seriously,’ on this view, entails not insertingad hoc exceptions to the theory’s laws for purposes of agreeing with the experimental data, as advocates of more traditional positivist or operationalist approaches, such as the famous Copenhagen Interpretation, are often accused of doing. Since much of the thesis concerns the bare formalism of quantum theory, and any successful interpreta- tion of quantum theory is likely to incorporate this formalism in some fashion or other, the skeptic about Everett and Bohm may still find a few pieces of pertinent material in the pages to follow. This material will be concentrated primarily in the sections pertaining to the Bare/Everett theory.
2.1.1
Motivation for Considering the Bare/Everett and
Bohm Theories in Parallel
One motivation for considering the Everett and Bohm theories together is that, if one is going to attempt to effect these reductions within the context of the Bohm theory, it is necessary anyway first to effect them within the context of the Bare/Everett theory (indeed, the project of this thesis grew out of my initial investigations into the classical domain of Bohm’s theory). The reason for this is that in the Bohm theory, the dynamics associated with the added structure of the theory - namely, the guidance equation for the added variables, designated ‘beables’ by John Bell (of the famous Bell Inequality), one the Bohm theory’s foremost proponents - depends on the value of the quantum state, but the value of the quantum state does not depend on the dynamics or configu- ration of the additional variables. Thus, in order to assess the behavior of the beables it is necessary, at least in a formal mathematical sense, to go through the Bare/Everett theory in determining the unitary evolution of the wave func- tion. Where Everettians and Bohmians disagree, primarily, is on whether the structure associated with the wave function is sufficient to save the appearances,
the theory to save the appearances.
Brown and Wallace have argued that Bohm’s theory is Everett’s theory ‘in denial,’ in the sense that Everett’s theory already contains all of the necessary mathematical structure to save the appearances, and so the additional con- figurations of Bohm’s theory are therefore merely epiphenomenal ‘idle wheels.’ However, their argument relies on the presumption that Everett’s theory does in- deed save the appearances - by no means a consensus opinion. Accepting Brown and Wallace’s point that, if Everett’s theory does indeed save the appearances, Bohm’s additional configurations are superfluous, I nevertheless maintain a con- sideration of Bohm’s theory out of recognition of the possibility that Everett’s theory may, for one reason or other, fail to save the appearances, and that Bohm’s additional configurations may offer the mechanism needed to address its shortcomings. The criticism that is currently the source of most informed skepticism about Everett’s theory is that it cannot adequately explain the role of probability - specifically, the success of the Born, or |ψ|2, Rule - in ordi-
nary quantum mechanics. Deutsch and Wallace have proposed a derivation of the Born Rule from the principles of rational decision theory, which has been notably defended by Greaves [28], [109], [42]. For further discussion of the ‘Ev- erett in denial’ charge against the Bohm theory, the reader should consult, in particular, the exchange between Brown/Wallace and Valentini [82], [103].
In section 2.3, I provide a brief summary of the accounts of measurement offered by the Bare/Everett and Bohm theories.